19.3.40 Linear operators

\( \underline{\text { Condition: }} \) a) Find the norm, spectrum and eigenvectors of the operator \( A x=M \cdot x \) in the space \( R^{2} \) with norm \( \|x\|_{p} \), where \( M \) is the given matrix. b) Determine for what numbers \( \alpha \) the mapping \( B \) of the form \( B x=\alpha \cdot M \cdot x+v \), where \( M \) is the given matrix, \( v \) is the given vector, is contractive in the space \( R^{2} \) with norm \( \|x\|_{p} \). c) Find its fixed point exactly, as well as approximately (by doing four iterations). \[ M=\left(\begin{array}{cc} 9 & 0 \\ -7 & 3 \end{array}\right), \quad v=\left(\begin{array}{l} 0 \\ 9 \end{array}\right), \quad p=\infty \text {. } \]